Question

(10 pts) Write a function to call python modules (either re-write or re-use the program you developed before) to calculate the derivative of In(1+x) at X = 1 using the forward difference, central difference and extrapolated dif- ference methods. Also use this function to create a similar log-log plot that compares the absolute value of the approximation error (you can ignore the rounding error) as a function of the size of the intervals used for these three methods. Choose the following 15 h values (10–1,10-2, ..., 10–15) for the x axis. Here you can use the actual mathematical function for the approximation error calculation. Also write down the x and y values for these points used to make this plot into a table like: h error(Forward) error(Central) error(Extrapolated) 10-1 10-2

Answer 1

https://trinket.io/python3/ea5cc4c872

import numpy as np
import matplotlib.pyplot as plt
def f(x):
return np.log(1+x);
h=np.logspace(-1,-15,15);
x=1;
Df1=(f(x+h)-f(x))/h;
Df2=(f(x+h)-f(x-h))/(2*h);
Df3=(f(x)-f(x-h))/h;
plt.loglog(h,abs(Df1-1.0/(1.0+x)));
plt.loglog(h,abs(Df2-1.0/(1.0+x)));
plt.loglog(h,abs(Df3-1.0/(1.0+x)));
plt.legend(['Forward','Central','Extrapolated']);
plt.show();
print(Df1)
print(Df2)
print(Df3)